Moment of Inertia

Note that in statics we are using the second moment of area as our moment of inertia. In dynamics a different moment of inertia is used (the mass moment of inertia). The differences between these two quantities are summarized in the table below:

	Mass moment of inertia	Area moment of inertia
Other names		Second moment of area
Description	Determines the torque needed to produce a desired angular rotation about an axis of rotation (resistance to rotation)	Determines the moment needed to produce a desired curvature about an axis(resistance to bending)
Equations	$$ I_{P,\hat{a}} = \iiint_\mathcal{B} \rho r^2 \,dV\ $$	$$ I_x = \int_{A}^{} y^2 \,dA \ $$ $$ I_y = \int_{A}^{} x^2 \,dA \ $$ $$ J_O = \int_{A}^{} r^2 \,dA = \int_{A}^{} (x^2+y^2) \,dA\ $$
Units	$ length*mass^2 $	$ length^4 $
Typical Equations	$$ \tau = I\alpha $$	$$ \sigma = (My)/I \ $$
Courses	TAM 212	TAM 210, TAM 251

Moment of Inertia (Second Moment of Area)

The moment of inertia used in statics is the "second moment of area", which is a mass property that determines the amount of torque that is needed to create an angular acceleration about a specific axis of rotation. The dimension of the area moment of inertia is $ length^4 $.\\ Moment of inertia about the x-axis:

$$ I_x = \int_{A}^{} y^2 \,dA \ $$

Moment of inertia about the y-axis:

$$ I_y = \int_{A}^{} x^2 \,dA \ $$

Polar moment of inertia:

$$ J_O = \int_{A}^{} r^2 \,dA = \int_{A}^{} (x^2+y^2) \,dA\ $$

Note that the polar moment of inertia depicts the measure of the distribution of area about a point (usually the origin), rather than about an axis.

Parallel axis theorem

Need to transcribe this to be the correct format and drawings

Fig: ParallelAxisThm

Taken from TAM 210 Lecture 3 - Slide 3

Combining moments of inertia

Need to transcribe this to be the correct format and drawings - show the example problem with the house-shaped object.

Fig: MOIComposite

Taken from TAM 210 Lecture 3 - Slide 3